"The criterion of ‘good design' of all electrical machines is expressed by a constant...which gives a measure of the effective utilization of...space for the transformation of energy." — Gabriel Kron, 1930 Abstract - After briefly highlighting the professional career of Dr. Eric Laithwaite, we turn to his intuitively obtained notion of "Goodness" as a physical parameter and demonstrate that it can be deduced from Maxwell's equations. Goodness is linked to the parallel concepts of Q and power factor, and related to machine efficiency. Examples of cosmic "Goodness" are discussed. Finally, negative resistance via regeneration and parametric regeneration is introduced as a practical means to accomplish "Hyper-Goodness." This paper has several goals. First we wish to illustrate the link between Eric Laithwaite's "Goodness factor," Gabriel Kron's criterion of good design (which measures the effective utilization of space), the circuit quality factor known as Q, and electrical power factor. Second, we provide a generalized derivation of Laithwaite's Goodness directly from Maxwell's equations. And, finally, we describe how parametric operation can be used to provide "Hyper-Goodness." (a) Who Was Eric Laithwaite? Dr. Laithwaite was well-known and his work has been highly respected in power engineering circles for a long, long time. In 1964 he was appointed as Professor of Heavy Electrical Engineering at London's Imperial College. In 1966 he was awarded the Royal Society's Medal for invention, and in 1967 he was appointed External Professor of Applied Electricity at The Royal Institution of Great Britain. He also taught at the University of Manchester Institute of Science & Technology (UMIST) and was a visiting Professor at Sussex University. He was a Life Fellow of the IEEE, a Fellow of the (British) IEE, and in 1986 he was awarded the IEEE's Tesla Medal. He started his career by making improvements to conventional electric motors, with some very clever designs including the spherical induction motor (which had very precise speed control using a mechanical lever mechanism, no electronics), and a homopolar induction motor (a self-starting, non-shaded pole, single solenoid winding motor, running off of single phase AC, quite an accomplishment without electronics). (b) Laithwaite and Maglev Historically, it was Laithwaite who performed the first serious MAGLEV research. He is best remembered for his invention (he always claimed re-invention, giving credit to Wheatstone) of the linear induction motor for high speed levitating trains as well as its variants the transverse flux motor (TFM) and the "magnetic river," a fully non-electronic (no semiconductors) magnetic levitation and propulsion system for high speed trains with over-damped motion about all three axes and strong centering forces to keep it on the track. He also made significant contributions to magnetic levitation, proving that an iron-backed aluminum levitator disc would float higher above its associated electromagnet and consume less power than if it had been a superconducting disc (room temperature electromagnet), hence he called it a hyperconductor. Towards the end of his life, Laithwaite was hired by NASA to design and supervise the construction of a small-scale model of an electromagnetic accelerator track for launching manned and unmanned payloads into orbit (the Marshall Space Flight Center (MSFC) MAGLIFTER Project), essentially eliminating the first stage of a three stage rocket. This track incorporated everything Laithwaite had learned over his lifetime of heavy electrical engineering design and included many counterintuitive details like very large air-gap surface windings, instead of the usual windings in slots, and a full magnetic river levitation and propulsion system (no superconductors). Two of the authors had the opportunity while working on a subcontract for the MAGLIFTER project out of MSFC (this was a few years after Laithwaite had passed away) to see the test track at the Redstone Arsenal. On the one hand it was a pleasure to see the culmination of a lifetime of Laithwaite's cunning handiwork in one place, and on the other it was sad that further research on this promising concept was later canceled. (c) Laithwaite's Publications Being an educator at heart, Laithwaite wrote numerous peer-reviewed journal articles, engineering textbooks, and even children's books on science. He also wrote a collection of articles for the magazine Electrical Review with the title "Electromagnetic Phenomena." Subtitles of the articles were: The Shape of an Electric Current; How the Low Speed Brushless AC Motor Was Invented; The Bigger the Better; The Smaller the Better; Rack and Pinion Motors; How an Experiment with Iron Filings Provided New Motor Design Information; The Key to the Cosmos; Mini-Magnets and Micro-Magnets; More About the Jumping Ring; The Parallel World: Is It Inevitable?; Poles or Circuits: A New Concept in Magnetism; Of Teapots and Electric Motors; The Evolution of a Three Dimensional Electric Motor; Electromagnetic Rivers for High Speed Transport; The Rules of Planet Earth; The Rules of Men; The Rules of God; The Strange Happening at Savoy Place; The One That Got Away; The Quick and the Dead; Superconductivity and the Midas Touch; The Hyperconductor and the Linomat; The Shape of Things to Come; The Multiplication of Bananas and Umbrellas; The Bigger They Are the Harder They Fall; 1975: A Space Odyssey; 2001: Learning in Parallel; The Impossible Dream; Electromagnetism Is a Whodunit; All Things Are Possible; The Continuing Story of Gyroscopic Magic. In these articles, Laithwaite used the forum to discuss what is wrong with engineering education, what is wrong with government funding of research, as well as giving details of his linear motor and magnetic levitation work, including the infamous "gyroscope episode." (d) Laithwaite and Gyroscopes In the mid-1970s, Laithwaite was asked to present a couple of lectures at the Royal Institution in London: one for the Friday Evening Discourses in 1973 and the other for the famous Faraday Christmas Lectures in 1974, where in both instances he demonstrated what appeared to be a reduction in the weight of a pair of counter rotating motorcycle wheels on a common shaft. When spun up with an electric drill, they could easily be lifted over his head with one hand. Later, in the BBC TV program series "Heretics" (which aired in the early 1980s), Laithwaite also stood on a scale and showed that his combined weight and the weight of the wheel decreased when the gyro wheel was spun up. He asserted that this appeared to contradict what Newton taught about the laws of motion. This resulted in his condemnation by the British scientific establishment—after which he lost tenure at Imperial College. (e) What Is the Goodness Factor? In 1965 Laithwaite derived (apparently by inspection) an equation for designing optimized electrical machines, which he called the "Goodness factor." This equation was, almost simultaneously, derived by several Russian researchers who called it the "Electromagnetic Reynolds Number." Even earlier, Gabriel Kron (back in 1930) had derived a similar equation for "good machine design." One reason that Laithwaite was able to make so much progress in electrical machine design was his ability to optimize the proportions of the machine for maximum performance in what would now be called a "genetic algorithm" based on his "Goodness factor" equation. Laithwaite's "genetic algorithm" was only derived for J x B machines. There are four Maxwell Stress Tensor machines: B^2 per unit area (magnetic machines); J x B machines (electromagnetic machines, such as conventional AC and DC motors and generators); rho*E machines such as the Franklin/Pogendorff/Jefimenko electrostatic machines; and Slepian/Heaviside d/dt (D x B) machines (the Maxwell Stress Tensor Space Drive). What we will show in this paper is that, instead of a derivation by inspection, one may formally deduce Goodness directly from Maxwell's equations. We start with basic questions. First, what is an electric circuit? Paris and Hurd define it: "A circuit is a configuration of electromagnetic structures which allow specialized fields to exist in confined regions of space." Lumped element circuit theory assumes no wave interference phenomena are present (current entering and leaving terminals is identical). What constitutes an electrical machine? Laithwaite defined it: "It might be defined as a device for transferring power from one location to another across space." He proceeds to sketch his fundamental machine as dual, multiply-connected topologies: "Viewed in the most fundamental way, every motor is no more than a magnetic circuit and an electric circuit linked together." The magnetic circuit must contain an air gap in any machine requiring mechanical movement (a transformer is an example with no moving parts). Laithwaite's fundamental machine links two interlocked loops. Laithwaite reasoned that the electric path was degraded by resistance: R = EMF / I = l_e / (sigma * A_e) = (rho * l_e) / A_e ohms (1) where l_e is the electrical path length, sigma is electrical conductivity, A_e is the electrical path cross-section, and rho is electrical resistivity. Similarly, the magnetic path is degraded by reluctance: Reluctance R_m = MMF / Flux = l_m / (mu * A_m) rels (2) where l_m is magnetic path length, mu is magnetic permeability, and A_m is cross-section. Force in a machine is proportional to J x B; therefore, the product of resistance and reluctance represents badness. Goodness is the reciprocal of badness: Goodness = 1 / Badness = 1 / (R * R_m) (3) To make the expression dimensionless, angular frequency omega is introduced for AC machines, yielding: Gamma = (1/R) * (1/R_m) * omega = (omega * mu * sigma * A_e * A_m) / (l_e * l_m) (4) This is Laithwaite's Goodness factor (using Gamma to avoid confusion with conductance G). If linear dimensions are doubled, goodness increases by a factor of four. (a) Problem Formulation Utilizing the time-harmonic Maxwell curl equations with e^(+jwt) time dependence: Curl(H) = (sigma + j*omega*epsilon) E = y_hat * E (5) Curl(E) = -j*omega*mu * H = -z_hat * H (6) where y_hat and z_hat are complex admittivity and impedivity. In integral form for uniform field distributions: Y = I / EMF = (sigma + j*omega*epsilon) * A_e / l_e (7) Z = V / MMF = j*omega*mu * A_m / l_m (8) A dimensionless electromagnetic parameter Gamma is defined by the scalar multiplication |Z * Y|: Gamma = | (V / MMF) * (I / EMF) | = | (j*omega*mu * (sigma + j*omega*epsilon) * A_e * A_m) / (l_e * l_m) | (12) (b) Passage to Conventional Machinery When conduction current dominates displacement current, Equation 12 reduces to Laithwaite's expression: Gamma -> (omega * mu * sigma * A_e * A_m) / (l_e * l_m) (13) In general: Z * Y = (j*omega*mu*sigma - omega^2*mu*epsilon) * (A_e * A_m) / (l_e * l_m) (14) With lumped parameters L = (mu * A_m)/l_m, C = (epsilon * A_e)/l_e, R = l_e / (sigma * A_e), LC = 1/omega_o^2, and Q = (omega_o * L) / R: Gamma = | -omega^2*L*C + j*(omega*L/R) | = | -(omega/omega_o)^2 + j*Q | (17) At power frequencies (omega << omega_o), Gamma reduces to Laithwaite's expression. For a coil-wound inductor with magnetic path l_m, area A_m, wire path l_e, area A_e, Ampere's law gives H = N*I / l_m and current density J = I / A_e. Power dissipation in copper is: P_d = 1/2 * (I^2 * l_e) / (sigma * A_e) (22) Total stored energy in the field is: W = 1/2 * L * I^2 = 1/2 * (mu * N^2 * A_m * I^2) / l_m (24) Using the energy definition of Q: Q = (2*pi * <energy stored>) / (energy lost per period) = (omega * W) / P_d = (mu * omega * sigma * A_e * A_m) / (l_e * l_m) = Gamma (27) Thus, Q = X_L / R approx Gamma (28). Goodness is the ratio of full-load current to no-load current to a first approximation. From Poynting's theorem, complex power is P_c = P_r +- j*P_x. In reactive circuits, the power factor (PF) is: PF = P_r / |P_c| = cos(theta) = R / |R +- jX| = 1 / |1 +- jQ| -> 1/Q (31) When R << X, PF -> 1/Q = R/|X| = 1/Gamma. The greater the Goodness, the smaller the machine losses and the lower the unloaded power factor. (a) Derivation Assuming losses are confined to the electric circuit (I^2*R) and reactive power is j*I^2*X: Efficiency eta = | 1 - (I^2*R) / (I^2*R + j*I^2*X) | = | Gamma / (Gamma - j) | = Gamma / sqrt(Gamma^2 + 1) (33) where Gamma = X/R. (b) Goodness Threshold: Gamma = 1 Second-order differential equations require Q >= 1/2 to possess oscillatory modes. For electromagnetic motors to develop sufficient starting/running torque, Goodness must exceed a threshold value. Tesla's single-phase induction motor failed to run for six years until 1888 when he added iron to the cross-axis, ensuring Gamma > 1. Goodness also applies to mechanical systems: a bouncing golf ball has a Q of ~35; a spinning top with rim mass has Goodness of 6 to 8. Gyroscopes require Gamma > 1 to exhibit true gyroscopic phenomena. Goodness characterizes cosmic and astrophysical phenomena: - Earth-Moon tidal dissipation gives an effective terrestrial Goodness Gamma = 1.2 x 10^12. - The Io-Jupiter unipolar dynamo creates an induced voltage of ~400 kV across Io, driving >5 million amperes along flux tubes connecting Io to Jupiter's ionosphere, yielding >10^12 W and a Goodness Gamma ~ 10^14. - Pulsar models (rotating neutron stars of ~1 solar mass spinning at 30 rev/sec and radiating 5.62 x 10^19 W) indicate a Goodness factor Gamma = 6.2 x 10^24. - Galactic dynamics and rotating Kerr black holes also fit within the framework of energy storage vs. dissipation. (a) Regenerative Radios By introducing negative resistance into a system, effective resistance decreases and Goodness rises beyond passive limits. Edwin H. Armstrong discovered the Principle of Regeneration in 1912, neutralizing positive resistance to raise effective Q and selectivity. Later developed into Q-multipliers by Harris and Villard, regeneration operates similarly in optical ring cavities. (b) Parametric Regeneration Leonid Mandelshtam and his school in the 1930s showed that periodic variation of a reactive parameter at 2*omega_o compensates for energy losses by injecting parametric negative resistance. Threshold for parametric oscillations is m > 2/Q = 2/Gamma (as in a child pumping a swing). If magnetic permeability mu or inductance L is parametrically modulated at twice the operating frequency, negative resistance is injected, enabling machines to operate in a 'Hyper-Goodness' regime (as demonstrated in Mandelshtam-Papalexi parametric generators and foreshadowed in Tesla's tuned machine patents). Laithwaite noted that while electromagnetic machines (induction, synchronous, commutator) improve as they get larger, magnetic machines (hysteresis, switched-reluctance, permanent magnet motors) improve as they get smaller. A unified equation for magnetic machines remains an open problem, potentially solvable using the remaining two Maxwell equations (divergence equations). Kron's criterion of good design and Laithwaite's machine Goodness factor can be directly derived from Maxwell's equations. Goodness is a fundamental metric-independent property describing systems with energy exchange and dissipation across engineering and cosmological scales. Parametric pumping allows injection of negative resistance to achieve Hyper-Goodness without violating thermodynamic laws. Goodness provides both a theoretical foundation and a practical tool for designing future high-efficiency electrical machinery. Dr. James F. Corum holds BSEE, MSEE, and Ph.D. degrees in Electrical Engineering from Ohio State University. He has served at the NSA, West Virginia University, Ohio Institute of Technology, Battelle Institute, SARA, ISR, and as CTO of National Electrodynamics. Kenneth L. Corum holds a B.A. in Physics from Gordon College and performed graduate work at UMASS. He has worked at Varian Associates, Pinzone Communications, and Sun Microsystems, publishing extensively on RF technology and antennas. Philip V. Pesavento holds a BSEE from Carnegie Mellon University with advanced studies at USC and UCLA, having worked at Hughes Aircraft, TRW, Henry Radio, SARA, ISR, Powercast, and National Electrodynamics.